Table Of ContentReply to the comment on:
“Does an atom interferometer test the gravitational redshift
at the Compton frequency?”
Peter Wolf,1 Luc Blanchet,2 Christian J. Bord´e,1,3 Serge
Reynaud,4 Christophe Salomon,5 and Claude Cohen-Tannoudji5
1LNE-SYRTE, CNRS UMR 8630, UPMC, Observatoire de Paris,
61 avenue de l’Observatoire, 75014 Paris, France
2GRECO, Institut d’Astrophysique de Paris, CNRS UMR 7095,
UPMC, 98bis boulevard Arago, 75014 Paris, France
3Laboratoire de Physique des Lasers, CNRS UMR 7538,
99 avenue Jean-Baptiste Cl´ement, 93430 Villetaneuse, France
4Laboratoire Kastler Brossel, CNRS UMR 8552, ENS, UPMC, Campus Jussieu, 75252 Paris, France
5Laboratoire Kastler Brossel et Coll`ege de France, CNRS UMR 8552,
2
ENS, UPMC, 24 rue Lhomond, 75231 Paris, France
1
(Dated: January 10, 2012)
0
2 Hohensee, Chu, Peters and Mu¨ller have submitted a comment (arXiv:1112.6039 [gr-qc]) on our
n paper “Does an atom interferometer test the gravitational redshift at the Compton frequency?”,
a Classical and Quantum Gravity 28, 145017 (2011), arXiv:1009.2485 [gr-qc]. Here we reply to this
J commentandshowthatthemainresultofourpaper,namelythatatominterferometricgravimeters
9 donot test thegravitational redshift at the Compton frequency,remains valid.
]
c In their comment Hohensee, Chu, Peters and Mu¨ller (referred to as HCPM here) [1] re-affirm their earlier inter-
q pretation, based on an analogy with classical clock experiments, of atom interferometric gravimeters as testing the
- gravitational redshift. They repeat previous arguments, in particular the cancellation between the time dilation and
r
g laser phase contributions to the phase shift, and arguments centered around Schiff’s conjecture. Here we re-affirm
[ that these arguments are incorrect or irrelevant, as shown below, and that the main result of our paper [2], namely
that atom interferometric gravimeters do not test the gravitationalredshift at Compton’s frequency, remains valid.
1
v First HCPMarguethat the absence of the Comptonfrequency (or equivalently the atom’s mass)in the final phase
8 shift, does not prevent the experiment from being a redshift test at that frequency. They claim that this is also the
7 case for redshift tests using standard clocks, like the Pound-Rebka experiment, where the redshift effect is cancelled
7 by a controlled first order Doppler effect, thus effectively leaving a null signal independent of the 14.4 keV transition
1
frequency. This analogyis fallacious: the Pound-Rebkaexperiment is basedona resonancebetweenatoms sitting at
.
1 the top and bottom of a tower. The controlled Doppler shift is just a clever experimental method for scanning the
0 resonancewhichisaffectedbytheredshift(cf. Figure1. in[3]). Theresultingsignalisthusnullontheexactcenterof
2
theresonancecurvebutnotatotherfrequencies. Intheatomgravimeterincontrast,thecancellationoftheCompton
1
frequency is inherent to the interferometer : the action phase shift involving the Compton frequency is always zero
:
v as shown in [2] and the measurement of that frequency (and of the associated redshift) is intrinsically impossible.
i Furthermore it is manifestly incorrect to state as HCPM do that the cancellation of the clock frequency is a generic
X
featureofanyredshiftmeasurement. Mostclockexperiments[4,5]measurefrequenciesoftwoelectromagneticsignals
r
a delivered by two clocks. The redshift effect is thus associated to a relation such as ∆f = f0∆U/c2 where ∆f is
the measured frequency difference and f the common proper frequency of the clocks. This could also be done in
0
principle for a Pound-Rebka-likeexperiment if experimentalmeans (sayfrequency combs)wereavailablefor accurate
measurements of frequencies in the X-ray domain. By contrast, in the atomic interferometer considered by HCPM,
there is no real emission of a photon at the Compton frequency and no measurement of the frequency of this photon
by a detector.
′
TheinterferometerconsideredbyHCPMisinfactnotanatomicclock. Thetwostatesgandg appearsymmetrically
in the two arms of the interferometer and no resonanceis obtainedwhen one varies the laser frequency. As explained
in [2], there are other non-symmetric atomic interferometers where the output signal exhibits resonances when the
laserfrequencyisvaried. Theyprovideameasurementoftheatomicfrequencyωgg′ (seediscussionafter(2.21)in[2]),
so that these interferometers can be also considered as atomic clocks oscillating at ωgg′. To get an oscillation at the
Compton frequency, one would need to have in the two arms of the interferometer two states, whose energies differ
by mc2, which is far beyond present day technology. Furthermore, to test the red shift at Compton frequency, two
different clocks of this type located at two different altitudes would have to be built and their frequencies compared.
In a related argument (eq. (1) and text below in their comment), HCPM restate their previous claim that in the
total phase shift the time dilation phase φ cancels the laser phase φ and not the gravitationalphase φ , whereas
t laser r
we have demonstrated in [2] that the free evolution phase φ = φ +φ is necessarily zero for all closed paths that
S r t
2
follow trajectories obtained from the same Lagrangianas the one used for the phase integral. The latter fact implies
that the difference of action integrals is zero and that the Compton frequency and redshift at that frequency are
unmeasurable, no matter what modified Lagrangian is used (i.e. independently of any redshift violating parameter
β). We re-affirm that the fundamental cancellation is between φ and φ , one reason being that φ = φ +φ = 0
r t S r t
is an invariant statement, since φ is the contour integral of the proper time (see first term of the right-hand side
S
of (1.3) in [2]) which is invariant under coordinate transformations. By contrast the statement that φ +φ = 0
t laser
is not invariant and is only true in the rest frame of the lasers. To see this in a simple example, analyze the atom
gravimeter in a frame that is freely falling in the Earth’s gravitational field. An elementary calculation shows that
φ and φ are separately zero in that frame, φ = φ = 0, but that φ = kgT2, where k is the laser wavevector
r t r t laser
and T the interrogationtime, because the lasersare acceleratedupwardsin that frame. Thus there is no cancellation
between φ and φ in that frame but still we have φ =φ +φ =0.
t laser S r t
Concerning Schiff’s conjecture, we do not agree with the analysis of HCPM. Schiff’s conjecture states that any
violation of the universality of free fall (UFF) also implies a violation of the universality of clock rates (UCR) and
viceversa. Doesthismeanthatitmakesnosensetodistinguishbetweenthe twotypesoftests? Ouransweris“No”,
because the relation between UFF and UCR tests depends on the experiments under consideration, on the nuclear
or atomic models used for the analysis of the clocks and the test masses, and on the particular model used for the
violation of UFF and UCR.
The next question is then whether the atom gravimetermeasurementis more directly comparable to UFF or UCR
tests ? As shown in [2] but also e.g. in [6] the answer is clearly UFF as it constrains the same parameters as other
UFFtests, whereasitrequiresnuclearandatomicmodelsto relateitto parametersmeasuredinUCRtests. Tomake
this point more explicit, let us quote the SME analysis presentedin [7] : the UFF test of [8] sets a limit on βBe−βAl
(to leading order), the atom gravimetertest sets a limit on βCs−βgrav (where “grav”stands for the test mass in the
classical gravimeter)and the Pound-Rebka experiment on ξMossb−βgrav. To relate the former two one only requires
57Fe
the atomic composition of Be, Al and grav (equ. (8) of [7]) but to relate them to the Pound-Rebka measurement one
additionally requires a nuclear model for the 14.4 keV transition in 57Fe (see text before (11) in [7]).
To conclude, let us briefly recall the compelling argument (see e.g. [6]) that atom interferometers test the UFF.
This argument is agreed on since the realizationof the first atomic gravimeters two decades ago [9]. The atoms with
inertial mass m and gravitational mass m obey the Lagrangian L = 1m z˙2−m gz in the gravitational field of
i g 2 i g
the Earth. The phase shift is the sum of the free evolution phase given by the difference of action integrals φ along
S
the two paths, and of the contribution φ coming from the interaction with the lasers. The free evolution phase
laser
φ is exactly zero for a closed total path (see e.g. [2]), and the phase shift reduces to φ = (m /m )gkT2 using
S laser g i
the position of the atoms deduced from the Lagrangian. Comparing with the measurement by a nearby classical
gravimeter of the gravitational acceleration of a freely falling macroscopic mass (corner cube), g = (M /M )g,
meas g i
one obtains to first order φ =(1+η)g kT2 where η =m /m −M /M is the Eo¨tvosparameter between the
laser meas g i g i
atoms and the corner cube, showing that measurement of the phase shift gives a test on possible violations of the
UFF between atoms and macroscopic masses.
[1] M.A.Hohensee,S.Chu,A.PetersandH.Mu¨ller,Commenton: “Doesanatominterferometertestthegravitationalredshift
at theCompton frequency?”, arXiv:1112.6039 [gr-qc].
[2] P. Wolf, L. Blanchet, Ch.J. Bord´e, S. Reynaud, C. Salomon, and C. Cohen-Tannoudji, Class. Quant. Grav. 28, 145017
(2011), arXiv:1009.2485 [gr-qc].
[3] R.V. Pound and J.L. Snider,Phys. Rev.140, B788 (1965).
[4] R. Vessot and M. Levine, Gen. Rel. and Grav. 10, 181 (1979).
[5] L. Cacciapuoti and C. Salomon, Eur. Phys.J. Spec. Top. 127, 57 (2009).
[6] D. Giulini (2011), arXiv:1105.0749 [gr-qc].
[7] M.A. Hohensee, S.Chu, A.Peters and H.Mu¨ller, Phys. Rev.Lett. 106, 151102 (2011).
[8] S. Schlamminger, K.-Y. Choi, T. Wagner, J. Gundlach, and E. Adelberger, Phys.Rev.Lett. 100, 041101 (2008).
[9] M. Kasevich and S.Chu, Phys.Rev.Lett. 67, 181 (1991).
